I was trying to work through a proof in Folland's book on Fourier analysis (Theorem 7.5c).

In his proof the conditions he gives on the function f are as follows:

- f is continuous
- f is piecwise smooth (i.e. f' is piecewise continuous)
- $\displaystyle f \in L^1(\mathbb{R})$
- $\displaystyle f' \in L^1(\mathbb{R})$

Then he writes, "Observe that since $\displaystyle f' \in L^1(\mathbb{R})$, the limit,

$\displaystyle \lim_{x\to \infty } f(x)=f(0)+ \int _0^{\infty }f'(x)dx$

exists, (I get this, simple application of the fundamental theorem of calculus),

and since $\displaystyle f \in L^1(\mathbb{R})$ this limit must be zero (this part, I dont get!)"